Obstacle on path does not interact with fluid
In case anybody else will need this, Export Animated Mesh under Physics tab -> Fluid worked for me when applied to the obstacle. In Blender commented "only use if really necessary", but it worked. One disadvantage here - it is not possible to set the Slip Type.

Maybe I just missed some settings, but without that I couldn't make it work. 

I also thought it may has something to do with the original position of the obstacle (before constrain to path added), which doesn't change, but this is just guess, not sure what is behind that in Blender's core.

This did lead me to the solution - Gimble answer



And here is the Blend file: